Full rank Cholesky factorization for rank deficient matrices

[EN] Let A be a rank deficient square matrix. We characterize the unique full rank Cholesky factorization LL^T of A where the factor L is a lower echelon matrix with positive leading entries. We compute an extended decomposition for the normal matrix B^TB where B is a rectangular rank deficient matrix. This decomposition is obtained without interchange of rows and without computing all entries of the normal matrix. Algorithms to compute both factorizations are given.